Abstract
We investigate collocation methods for the efficient solution of singular boundary value problems with an essential singularity. We give numerical evidence that this approach indeed yields high order solutions. Moreover, we discuss the issue of a posteriori error estimation for the collocation solution. An estimate based on the defect correction principle, which has been successfully applied to problems with a singularity of the first kind, is less robust with respect to an essential singularity than a classical strategy based on mesh halving.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Auzinger, W., Kneisl, G., Koch, O., Weinmüller, E.: A collocation code for boundary value problems in ordinary differential equations. Numer. Algorithms (to appear), Also available as ANUM Preprint Nr. 18/01 at http://www.math.tuwien.ac.at/~inst115/preprints.htm
Auzinger, W., Koch, O., Petrickovic, J., Weinmüller, E.: The numerical solution of boundary value problems with an essential singularity. Techn. Rep. ANUM Preprint Nr. 3/03, Inst. for Appl. Math. and Numer. Anal., Vienna Univ. of Technology, Austria (2003), Available at http://www.math.tuwien.ac.at/~inst115/preprints.htm
Auzinger, W., Koch, O., Weinmüller, E.: Efficient collocation schemes for singular boundary value problems. Numer. Algorithms 31, 5–25 (2002)
Auzinger, W., Koch, O., Weinmüller, E.: Analysis of a new error estimate for collocation methods applied to singular boundary value problems. SIAM J. Numer. Anal. (submitted to), Also available as ANUM Preprint Nr. 13/02 at http://www.math.tuwien.ac.at/~inst115/preprints.htm
Bergström, A.: Electromagnetic theory of strong interaction. Phys. Rev. D 8, 4394–4402 (1973)
Boor, C.D., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal. 10, 582–606 (1973)
Hoog, F.D., Weiss, R.: The numerical solution of boundary value problems with an essential singularity. SIAM J. Numer. Anal. 16, 637–669 (1979)
Hoog, F.D., Weiss, R.: On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind. SIAM J. Math. Anal. 11, 41–60 (1980)
Lentini, M., Keller, H.: Boundary value problems on semi-infinite intervals and their numerical solution. SIAM J. Numer. Anal. 17, 577–604 (1980)
Markowich, P.: Asymptotic analysis of von Karman flows. SIAM J. Appl. Math. 42, 549–557 (1982)
Schlichting, H.: Boundary Layer Theory. McGraw-Hill, New York (1968)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Auzinger, W., Koch, O., Weinmüller, E. (2004). Collocation Methods for Boundary Value Problems with an Essential Singularity. In: Lirkov, I., Margenov, S., Waśniewski, J., Yalamov, P. (eds) Large-Scale Scientific Computing. LSSC 2003. Lecture Notes in Computer Science, vol 2907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24588-9_39
Download citation
DOI: https://doi.org/10.1007/978-3-540-24588-9_39
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21090-0
Online ISBN: 978-3-540-24588-9
eBook Packages: Springer Book Archive